Supervised by Professor Luke Postle, a faculty member with the Department of Combinatorics and Optimization, Kelly defended his thesis, His thesis, , on July 16, 2019.
Kelly鈥檚 thesis made significant contributions to long-standing open problems, including the launch of two major paradigms in graph coloring 鈥 a research area for more than 100 years. The study of graph coloring focuses on the following problem: a collection of objects needs a color assigned to each one to satisfy some basic rules such as certain pairs are not allowed to receive the same color. The mathematician鈥檚 task is to find a color assignment using as few colors as possible. One application of the use of graph coloring is the correspondence of colors to time slots to model a scheduling problem.
鈥淚 believe the results of Tom鈥檚 thesis are of the highest caliber and that they will leave a lasting impact on the field,鈥 said Postle. 鈥淗e is an excellent mathematician with an eye for good problems, more than that for field-changing paradigms.鈥
Kelly was able to prove a unifying theorem based on Johansson鈥檚 results from the 1990s while also providing a new bound on the chromatic number. He also strengthens one of Reed鈥檚 theorems and made progress toward Hadwiger鈥檚 Conjecture. Finally, he developed a new theory that he called 鈥渇ractional coloring with local demands鈥 which connected classical graph coloring and extremal graph theory, providing a more robust version of Turan鈥檚 Theorem which improves the Caro-Wei bound on the independence number.
鈥淯ltimately, I was surprised by how deep this theory turned out to be,鈥 said Kelly. 鈥淭here were unexpected connections to other areas of combinatorics, and there are still many opportunities for more research to be done.鈥
Kelly came to the University of 蓝莓视频 as a Master鈥檚 student in 2015 after completing his undergraduate degree at Princeton University. He was quickly transferred directly into the PhD program.
Kelly is currently a postdoctoral researcher at the University of Birmingham in the United Kingdom.